Carlyle circle
circle in a coordinate plane associated with a quadratic equation
In mathematics, a Carlyle circle is a certain circle in a coordinate plane associated with a quadratic equation; it is named after Thomas Carlyle. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons.
Definition
Given a quadratic equation in the form
x2 − sx + p = 0
the circle associated to it in the coordinate plane having the line segment joining the points A(0, 1) and B(s, p) as a diameter is called the Carlyle circle of the quadratic equation.
Defining property
The defining property of the Carlyle circle can be established thus: the equation of the circle having the line segment AB as diameter is
x(x − s) + (y − 1)(y − p) = 0.
The abscissas of the points where the circle intersects the x-axis are the roots of the equation (obtained by setting y = 0 in the equation of the circle)
x2 − sx + p = 0.
Construction of regular polygons
Regular pentagon
The problem of constructing a regular pentagon is equivalent to the problem of constructing the roots of the equation
z5 − 1 = 0.
One root of this equation is z0 = 1 which corresponds to the point P0(1, 0). Removing the factor corresponding to this root, the other roots turn out to be roots of the equation
z4 + z3 + z2 + z + 1 = 0.
These roots can be represented in the form ω, ω2, ω3, ω4 where ω = exp (2iπ/5).
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